Properties

Label 6003.2.a.n.1.12
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 54 x^{8} - 188 x^{7} - 77 x^{6} + 342 x^{5} + 13 x^{4} - 215 x^{3} + 9 x^{2} + 37 x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(2.66955\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.66955 q^{2} +5.12647 q^{4} +2.89555 q^{5} -0.268248 q^{7} +8.34627 q^{8} +O(q^{10})\) \(q+2.66955 q^{2} +5.12647 q^{4} +2.89555 q^{5} -0.268248 q^{7} +8.34627 q^{8} +7.72982 q^{10} -5.35238 q^{11} +3.85670 q^{13} -0.716099 q^{14} +12.0278 q^{16} +3.41249 q^{17} -4.28684 q^{19} +14.8440 q^{20} -14.2884 q^{22} +1.00000 q^{23} +3.38424 q^{25} +10.2956 q^{26} -1.37516 q^{28} +1.00000 q^{29} +3.06011 q^{31} +15.4162 q^{32} +9.10980 q^{34} -0.776726 q^{35} +5.65787 q^{37} -11.4439 q^{38} +24.1671 q^{40} +2.62648 q^{41} -6.20369 q^{43} -27.4389 q^{44} +2.66955 q^{46} +12.2344 q^{47} -6.92804 q^{49} +9.03438 q^{50} +19.7713 q^{52} +12.5555 q^{53} -15.4981 q^{55} -2.23887 q^{56} +2.66955 q^{58} -9.30093 q^{59} +9.39308 q^{61} +8.16910 q^{62} +17.0987 q^{64} +11.1673 q^{65} -5.45335 q^{67} +17.4940 q^{68} -2.07350 q^{70} +14.0578 q^{71} -6.27709 q^{73} +15.1039 q^{74} -21.9764 q^{76} +1.43576 q^{77} -9.39080 q^{79} +34.8271 q^{80} +7.01152 q^{82} -4.93812 q^{83} +9.88105 q^{85} -16.5610 q^{86} -44.6724 q^{88} -6.26188 q^{89} -1.03455 q^{91} +5.12647 q^{92} +32.6602 q^{94} -12.4128 q^{95} -3.54652 q^{97} -18.4947 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8} + 6 q^{11} - 15 q^{13} + 8 q^{14} + 17 q^{16} + 18 q^{17} - 6 q^{19} + 39 q^{20} - 5 q^{22} + 12 q^{23} + 14 q^{25} + 3 q^{26} - 19 q^{28} + 12 q^{29} + 16 q^{31} + 21 q^{32} - 7 q^{34} + 11 q^{35} - q^{37} + 24 q^{38} + 30 q^{40} - 3 q^{41} - 23 q^{43} - 23 q^{44} + 3 q^{46} + 35 q^{47} + 3 q^{49} + 2 q^{50} + 45 q^{53} + 17 q^{55} + 17 q^{56} + 3 q^{58} + 11 q^{59} + 4 q^{61} + 7 q^{62} + 15 q^{64} - 5 q^{65} - 19 q^{67} - q^{68} + 14 q^{70} - 19 q^{71} + 10 q^{73} + 15 q^{74} - 4 q^{76} + 39 q^{77} + 17 q^{79} + 90 q^{80} - 3 q^{82} + 12 q^{83} + 14 q^{85} - 17 q^{86} - 2 q^{88} + 20 q^{89} + 11 q^{91} + 11 q^{92} + 13 q^{94} - 12 q^{95} - 12 q^{97} - 75 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.66955 1.88765 0.943827 0.330440i \(-0.107197\pi\)
0.943827 + 0.330440i \(0.107197\pi\)
\(3\) 0 0
\(4\) 5.12647 2.56324
\(5\) 2.89555 1.29493 0.647466 0.762095i \(-0.275829\pi\)
0.647466 + 0.762095i \(0.275829\pi\)
\(6\) 0 0
\(7\) −0.268248 −0.101388 −0.0506940 0.998714i \(-0.516143\pi\)
−0.0506940 + 0.998714i \(0.516143\pi\)
\(8\) 8.34627 2.95085
\(9\) 0 0
\(10\) 7.72982 2.44438
\(11\) −5.35238 −1.61380 −0.806902 0.590685i \(-0.798857\pi\)
−0.806902 + 0.590685i \(0.798857\pi\)
\(12\) 0 0
\(13\) 3.85670 1.06966 0.534828 0.844961i \(-0.320376\pi\)
0.534828 + 0.844961i \(0.320376\pi\)
\(14\) −0.716099 −0.191386
\(15\) 0 0
\(16\) 12.0278 3.00695
\(17\) 3.41249 0.827650 0.413825 0.910356i \(-0.364192\pi\)
0.413825 + 0.910356i \(0.364192\pi\)
\(18\) 0 0
\(19\) −4.28684 −0.983468 −0.491734 0.870746i \(-0.663637\pi\)
−0.491734 + 0.870746i \(0.663637\pi\)
\(20\) 14.8440 3.31922
\(21\) 0 0
\(22\) −14.2884 −3.04630
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 3.38424 0.676848
\(26\) 10.2956 2.01914
\(27\) 0 0
\(28\) −1.37516 −0.259882
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 3.06011 0.549612 0.274806 0.961500i \(-0.411386\pi\)
0.274806 + 0.961500i \(0.411386\pi\)
\(32\) 15.4162 2.72523
\(33\) 0 0
\(34\) 9.10980 1.56232
\(35\) −0.776726 −0.131291
\(36\) 0 0
\(37\) 5.65787 0.930147 0.465074 0.885272i \(-0.346028\pi\)
0.465074 + 0.885272i \(0.346028\pi\)
\(38\) −11.4439 −1.85645
\(39\) 0 0
\(40\) 24.1671 3.82115
\(41\) 2.62648 0.410188 0.205094 0.978742i \(-0.434250\pi\)
0.205094 + 0.978742i \(0.434250\pi\)
\(42\) 0 0
\(43\) −6.20369 −0.946054 −0.473027 0.881048i \(-0.656839\pi\)
−0.473027 + 0.881048i \(0.656839\pi\)
\(44\) −27.4389 −4.13656
\(45\) 0 0
\(46\) 2.66955 0.393603
\(47\) 12.2344 1.78456 0.892282 0.451478i \(-0.149103\pi\)
0.892282 + 0.451478i \(0.149103\pi\)
\(48\) 0 0
\(49\) −6.92804 −0.989720
\(50\) 9.03438 1.27765
\(51\) 0 0
\(52\) 19.7713 2.74178
\(53\) 12.5555 1.72463 0.862316 0.506371i \(-0.169013\pi\)
0.862316 + 0.506371i \(0.169013\pi\)
\(54\) 0 0
\(55\) −15.4981 −2.08977
\(56\) −2.23887 −0.299181
\(57\) 0 0
\(58\) 2.66955 0.350529
\(59\) −9.30093 −1.21088 −0.605439 0.795892i \(-0.707002\pi\)
−0.605439 + 0.795892i \(0.707002\pi\)
\(60\) 0 0
\(61\) 9.39308 1.20266 0.601330 0.799001i \(-0.294638\pi\)
0.601330 + 0.799001i \(0.294638\pi\)
\(62\) 8.16910 1.03748
\(63\) 0 0
\(64\) 17.0987 2.13734
\(65\) 11.1673 1.38513
\(66\) 0 0
\(67\) −5.45335 −0.666233 −0.333117 0.942886i \(-0.608100\pi\)
−0.333117 + 0.942886i \(0.608100\pi\)
\(68\) 17.4940 2.12146
\(69\) 0 0
\(70\) −2.07350 −0.247831
\(71\) 14.0578 1.66835 0.834175 0.551500i \(-0.185944\pi\)
0.834175 + 0.551500i \(0.185944\pi\)
\(72\) 0 0
\(73\) −6.27709 −0.734677 −0.367339 0.930087i \(-0.619731\pi\)
−0.367339 + 0.930087i \(0.619731\pi\)
\(74\) 15.1039 1.75580
\(75\) 0 0
\(76\) −21.9764 −2.52086
\(77\) 1.43576 0.163620
\(78\) 0 0
\(79\) −9.39080 −1.05655 −0.528274 0.849074i \(-0.677160\pi\)
−0.528274 + 0.849074i \(0.677160\pi\)
\(80\) 34.8271 3.89379
\(81\) 0 0
\(82\) 7.01152 0.774293
\(83\) −4.93812 −0.542029 −0.271014 0.962575i \(-0.587359\pi\)
−0.271014 + 0.962575i \(0.587359\pi\)
\(84\) 0 0
\(85\) 9.88105 1.07175
\(86\) −16.5610 −1.78582
\(87\) 0 0
\(88\) −44.6724 −4.76209
\(89\) −6.26188 −0.663758 −0.331879 0.943322i \(-0.607683\pi\)
−0.331879 + 0.943322i \(0.607683\pi\)
\(90\) 0 0
\(91\) −1.03455 −0.108450
\(92\) 5.12647 0.534472
\(93\) 0 0
\(94\) 32.6602 3.36864
\(95\) −12.4128 −1.27352
\(96\) 0 0
\(97\) −3.54652 −0.360094 −0.180047 0.983658i \(-0.557625\pi\)
−0.180047 + 0.983658i \(0.557625\pi\)
\(98\) −18.4947 −1.86825
\(99\) 0 0
\(100\) 17.3492 1.73492
\(101\) −1.47360 −0.146629 −0.0733143 0.997309i \(-0.523358\pi\)
−0.0733143 + 0.997309i \(0.523358\pi\)
\(102\) 0 0
\(103\) −16.0201 −1.57851 −0.789256 0.614065i \(-0.789533\pi\)
−0.789256 + 0.614065i \(0.789533\pi\)
\(104\) 32.1890 3.15639
\(105\) 0 0
\(106\) 33.5175 3.25551
\(107\) 16.6653 1.61109 0.805546 0.592534i \(-0.201872\pi\)
0.805546 + 0.592534i \(0.201872\pi\)
\(108\) 0 0
\(109\) −13.9091 −1.33225 −0.666124 0.745841i \(-0.732048\pi\)
−0.666124 + 0.745841i \(0.732048\pi\)
\(110\) −41.3729 −3.94475
\(111\) 0 0
\(112\) −3.22643 −0.304869
\(113\) 0.244699 0.0230193 0.0115096 0.999934i \(-0.496336\pi\)
0.0115096 + 0.999934i \(0.496336\pi\)
\(114\) 0 0
\(115\) 2.89555 0.270012
\(116\) 5.12647 0.475981
\(117\) 0 0
\(118\) −24.8293 −2.28572
\(119\) −0.915392 −0.0839139
\(120\) 0 0
\(121\) 17.6480 1.60436
\(122\) 25.0753 2.27021
\(123\) 0 0
\(124\) 15.6876 1.40879
\(125\) −4.67853 −0.418460
\(126\) 0 0
\(127\) 1.70418 0.151221 0.0756107 0.997137i \(-0.475909\pi\)
0.0756107 + 0.997137i \(0.475909\pi\)
\(128\) 14.8133 1.30932
\(129\) 0 0
\(130\) 29.8116 2.61465
\(131\) −10.1176 −0.883976 −0.441988 0.897021i \(-0.645727\pi\)
−0.441988 + 0.897021i \(0.645727\pi\)
\(132\) 0 0
\(133\) 1.14993 0.0997119
\(134\) −14.5580 −1.25762
\(135\) 0 0
\(136\) 28.4816 2.44227
\(137\) −2.99872 −0.256198 −0.128099 0.991761i \(-0.540888\pi\)
−0.128099 + 0.991761i \(0.540888\pi\)
\(138\) 0 0
\(139\) −11.8537 −1.00541 −0.502707 0.864457i \(-0.667662\pi\)
−0.502707 + 0.864457i \(0.667662\pi\)
\(140\) −3.98186 −0.336529
\(141\) 0 0
\(142\) 37.5278 3.14927
\(143\) −20.6425 −1.72621
\(144\) 0 0
\(145\) 2.89555 0.240463
\(146\) −16.7570 −1.38682
\(147\) 0 0
\(148\) 29.0049 2.38419
\(149\) 1.41163 0.115645 0.0578227 0.998327i \(-0.481584\pi\)
0.0578227 + 0.998327i \(0.481584\pi\)
\(150\) 0 0
\(151\) 4.16319 0.338795 0.169398 0.985548i \(-0.445818\pi\)
0.169398 + 0.985548i \(0.445818\pi\)
\(152\) −35.7791 −2.90207
\(153\) 0 0
\(154\) 3.83284 0.308859
\(155\) 8.86071 0.711710
\(156\) 0 0
\(157\) −17.8086 −1.42128 −0.710641 0.703555i \(-0.751595\pi\)
−0.710641 + 0.703555i \(0.751595\pi\)
\(158\) −25.0692 −1.99440
\(159\) 0 0
\(160\) 44.6385 3.52898
\(161\) −0.268248 −0.0211409
\(162\) 0 0
\(163\) 12.3775 0.969482 0.484741 0.874658i \(-0.338914\pi\)
0.484741 + 0.874658i \(0.338914\pi\)
\(164\) 13.4646 1.05141
\(165\) 0 0
\(166\) −13.1825 −1.02316
\(167\) 6.88064 0.532440 0.266220 0.963912i \(-0.414225\pi\)
0.266220 + 0.963912i \(0.414225\pi\)
\(168\) 0 0
\(169\) 1.87412 0.144163
\(170\) 26.3779 2.02309
\(171\) 0 0
\(172\) −31.8031 −2.42496
\(173\) 6.22503 0.473280 0.236640 0.971597i \(-0.423954\pi\)
0.236640 + 0.971597i \(0.423954\pi\)
\(174\) 0 0
\(175\) −0.907814 −0.0686243
\(176\) −64.3773 −4.85262
\(177\) 0 0
\(178\) −16.7164 −1.25295
\(179\) −22.4493 −1.67794 −0.838969 0.544179i \(-0.816841\pi\)
−0.838969 + 0.544179i \(0.816841\pi\)
\(180\) 0 0
\(181\) −4.29095 −0.318944 −0.159472 0.987202i \(-0.550979\pi\)
−0.159472 + 0.987202i \(0.550979\pi\)
\(182\) −2.76178 −0.204717
\(183\) 0 0
\(184\) 8.34627 0.615295
\(185\) 16.3827 1.20448
\(186\) 0 0
\(187\) −18.2649 −1.33567
\(188\) 62.7191 4.57426
\(189\) 0 0
\(190\) −33.1365 −2.40397
\(191\) −16.7358 −1.21096 −0.605479 0.795861i \(-0.707018\pi\)
−0.605479 + 0.795861i \(0.707018\pi\)
\(192\) 0 0
\(193\) 16.2950 1.17294 0.586468 0.809972i \(-0.300518\pi\)
0.586468 + 0.809972i \(0.300518\pi\)
\(194\) −9.46759 −0.679733
\(195\) 0 0
\(196\) −35.5164 −2.53689
\(197\) −20.6304 −1.46986 −0.734928 0.678145i \(-0.762784\pi\)
−0.734928 + 0.678145i \(0.762784\pi\)
\(198\) 0 0
\(199\) −12.8346 −0.909822 −0.454911 0.890537i \(-0.650329\pi\)
−0.454911 + 0.890537i \(0.650329\pi\)
\(200\) 28.2458 1.99728
\(201\) 0 0
\(202\) −3.93384 −0.276784
\(203\) −0.268248 −0.0188273
\(204\) 0 0
\(205\) 7.60513 0.531165
\(206\) −42.7665 −2.97968
\(207\) 0 0
\(208\) 46.3876 3.21640
\(209\) 22.9448 1.58712
\(210\) 0 0
\(211\) 9.39689 0.646908 0.323454 0.946244i \(-0.395156\pi\)
0.323454 + 0.946244i \(0.395156\pi\)
\(212\) 64.3655 4.42064
\(213\) 0 0
\(214\) 44.4887 3.04118
\(215\) −17.9631 −1.22507
\(216\) 0 0
\(217\) −0.820867 −0.0557241
\(218\) −37.1309 −2.51482
\(219\) 0 0
\(220\) −79.4507 −5.35656
\(221\) 13.1609 0.885301
\(222\) 0 0
\(223\) −21.7757 −1.45821 −0.729104 0.684403i \(-0.760063\pi\)
−0.729104 + 0.684403i \(0.760063\pi\)
\(224\) −4.13536 −0.276305
\(225\) 0 0
\(226\) 0.653234 0.0434525
\(227\) −10.6634 −0.707756 −0.353878 0.935292i \(-0.615137\pi\)
−0.353878 + 0.935292i \(0.615137\pi\)
\(228\) 0 0
\(229\) 9.68682 0.640123 0.320062 0.947397i \(-0.396296\pi\)
0.320062 + 0.947397i \(0.396296\pi\)
\(230\) 7.72982 0.509689
\(231\) 0 0
\(232\) 8.34627 0.547959
\(233\) −17.9188 −1.17390 −0.586949 0.809624i \(-0.699671\pi\)
−0.586949 + 0.809624i \(0.699671\pi\)
\(234\) 0 0
\(235\) 35.4253 2.31089
\(236\) −47.6810 −3.10377
\(237\) 0 0
\(238\) −2.44368 −0.158400
\(239\) 1.53321 0.0991752 0.0495876 0.998770i \(-0.484209\pi\)
0.0495876 + 0.998770i \(0.484209\pi\)
\(240\) 0 0
\(241\) 5.97835 0.385099 0.192550 0.981287i \(-0.438324\pi\)
0.192550 + 0.981287i \(0.438324\pi\)
\(242\) 47.1121 3.02848
\(243\) 0 0
\(244\) 48.1534 3.08270
\(245\) −20.0605 −1.28162
\(246\) 0 0
\(247\) −16.5330 −1.05197
\(248\) 25.5405 1.62182
\(249\) 0 0
\(250\) −12.4895 −0.789908
\(251\) −6.57893 −0.415258 −0.207629 0.978208i \(-0.566575\pi\)
−0.207629 + 0.978208i \(0.566575\pi\)
\(252\) 0 0
\(253\) −5.35238 −0.336501
\(254\) 4.54938 0.285454
\(255\) 0 0
\(256\) 5.34744 0.334215
\(257\) −2.86807 −0.178905 −0.0894527 0.995991i \(-0.528512\pi\)
−0.0894527 + 0.995991i \(0.528512\pi\)
\(258\) 0 0
\(259\) −1.51771 −0.0943058
\(260\) 57.2488 3.55042
\(261\) 0 0
\(262\) −27.0093 −1.66864
\(263\) −16.9954 −1.04798 −0.523991 0.851724i \(-0.675558\pi\)
−0.523991 + 0.851724i \(0.675558\pi\)
\(264\) 0 0
\(265\) 36.3552 2.23328
\(266\) 3.06980 0.188222
\(267\) 0 0
\(268\) −27.9565 −1.70771
\(269\) 20.5857 1.25514 0.627568 0.778562i \(-0.284051\pi\)
0.627568 + 0.778562i \(0.284051\pi\)
\(270\) 0 0
\(271\) 26.1461 1.58826 0.794130 0.607747i \(-0.207927\pi\)
0.794130 + 0.607747i \(0.207927\pi\)
\(272\) 41.0447 2.48870
\(273\) 0 0
\(274\) −8.00522 −0.483613
\(275\) −18.1137 −1.09230
\(276\) 0 0
\(277\) −5.31984 −0.319638 −0.159819 0.987146i \(-0.551091\pi\)
−0.159819 + 0.987146i \(0.551091\pi\)
\(278\) −31.6439 −1.89787
\(279\) 0 0
\(280\) −6.48276 −0.387419
\(281\) 10.8523 0.647396 0.323698 0.946160i \(-0.395074\pi\)
0.323698 + 0.946160i \(0.395074\pi\)
\(282\) 0 0
\(283\) −6.31750 −0.375536 −0.187768 0.982213i \(-0.560125\pi\)
−0.187768 + 0.982213i \(0.560125\pi\)
\(284\) 72.0668 4.27637
\(285\) 0 0
\(286\) −55.1062 −3.25850
\(287\) −0.704548 −0.0415881
\(288\) 0 0
\(289\) −5.35491 −0.314995
\(290\) 7.72982 0.453910
\(291\) 0 0
\(292\) −32.1793 −1.88315
\(293\) 3.14022 0.183454 0.0917268 0.995784i \(-0.470761\pi\)
0.0917268 + 0.995784i \(0.470761\pi\)
\(294\) 0 0
\(295\) −26.9313 −1.56800
\(296\) 47.2221 2.74473
\(297\) 0 0
\(298\) 3.76842 0.218298
\(299\) 3.85670 0.223039
\(300\) 0 0
\(301\) 1.66412 0.0959185
\(302\) 11.1138 0.639528
\(303\) 0 0
\(304\) −51.5612 −2.95724
\(305\) 27.1982 1.55736
\(306\) 0 0
\(307\) 19.7708 1.12838 0.564191 0.825644i \(-0.309188\pi\)
0.564191 + 0.825644i \(0.309188\pi\)
\(308\) 7.36040 0.419398
\(309\) 0 0
\(310\) 23.6541 1.34346
\(311\) 15.9208 0.902788 0.451394 0.892325i \(-0.350927\pi\)
0.451394 + 0.892325i \(0.350927\pi\)
\(312\) 0 0
\(313\) −32.2341 −1.82198 −0.910991 0.412427i \(-0.864681\pi\)
−0.910991 + 0.412427i \(0.864681\pi\)
\(314\) −47.5409 −2.68289
\(315\) 0 0
\(316\) −48.1417 −2.70818
\(317\) 4.03526 0.226643 0.113321 0.993558i \(-0.463851\pi\)
0.113321 + 0.993558i \(0.463851\pi\)
\(318\) 0 0
\(319\) −5.35238 −0.299676
\(320\) 49.5102 2.76770
\(321\) 0 0
\(322\) −0.716099 −0.0399066
\(323\) −14.6288 −0.813967
\(324\) 0 0
\(325\) 13.0520 0.723994
\(326\) 33.0423 1.83005
\(327\) 0 0
\(328\) 21.9213 1.21040
\(329\) −3.28184 −0.180934
\(330\) 0 0
\(331\) 17.2270 0.946880 0.473440 0.880826i \(-0.343012\pi\)
0.473440 + 0.880826i \(0.343012\pi\)
\(332\) −25.3151 −1.38935
\(333\) 0 0
\(334\) 18.3682 1.00506
\(335\) −15.7905 −0.862726
\(336\) 0 0
\(337\) −5.67028 −0.308880 −0.154440 0.988002i \(-0.549357\pi\)
−0.154440 + 0.988002i \(0.549357\pi\)
\(338\) 5.00305 0.272130
\(339\) 0 0
\(340\) 50.6550 2.74715
\(341\) −16.3789 −0.886966
\(342\) 0 0
\(343\) 3.73616 0.201734
\(344\) −51.7776 −2.79166
\(345\) 0 0
\(346\) 16.6180 0.893390
\(347\) −11.3415 −0.608843 −0.304422 0.952537i \(-0.598463\pi\)
−0.304422 + 0.952537i \(0.598463\pi\)
\(348\) 0 0
\(349\) −30.3888 −1.62668 −0.813338 0.581792i \(-0.802352\pi\)
−0.813338 + 0.581792i \(0.802352\pi\)
\(350\) −2.42345 −0.129539
\(351\) 0 0
\(352\) −82.5134 −4.39798
\(353\) −32.2636 −1.71722 −0.858609 0.512631i \(-0.828671\pi\)
−0.858609 + 0.512631i \(0.828671\pi\)
\(354\) 0 0
\(355\) 40.7050 2.16040
\(356\) −32.1014 −1.70137
\(357\) 0 0
\(358\) −59.9294 −3.16737
\(359\) −2.03999 −0.107667 −0.0538333 0.998550i \(-0.517144\pi\)
−0.0538333 + 0.998550i \(0.517144\pi\)
\(360\) 0 0
\(361\) −0.623032 −0.0327911
\(362\) −11.4549 −0.602056
\(363\) 0 0
\(364\) −5.30359 −0.277984
\(365\) −18.1756 −0.951357
\(366\) 0 0
\(367\) −25.1042 −1.31043 −0.655214 0.755443i \(-0.727422\pi\)
−0.655214 + 0.755443i \(0.727422\pi\)
\(368\) 12.0278 0.626992
\(369\) 0 0
\(370\) 43.7343 2.27364
\(371\) −3.36799 −0.174857
\(372\) 0 0
\(373\) 22.9976 1.19077 0.595386 0.803439i \(-0.296999\pi\)
0.595386 + 0.803439i \(0.296999\pi\)
\(374\) −48.7591 −2.52127
\(375\) 0 0
\(376\) 102.111 5.26598
\(377\) 3.85670 0.198630
\(378\) 0 0
\(379\) 29.9225 1.53702 0.768509 0.639839i \(-0.220999\pi\)
0.768509 + 0.639839i \(0.220999\pi\)
\(380\) −63.6338 −3.26434
\(381\) 0 0
\(382\) −44.6769 −2.28587
\(383\) 21.5495 1.10113 0.550563 0.834793i \(-0.314413\pi\)
0.550563 + 0.834793i \(0.314413\pi\)
\(384\) 0 0
\(385\) 4.15733 0.211877
\(386\) 43.5001 2.21410
\(387\) 0 0
\(388\) −18.1811 −0.923007
\(389\) 18.0550 0.915424 0.457712 0.889101i \(-0.348669\pi\)
0.457712 + 0.889101i \(0.348669\pi\)
\(390\) 0 0
\(391\) 3.41249 0.172577
\(392\) −57.8233 −2.92052
\(393\) 0 0
\(394\) −55.0738 −2.77458
\(395\) −27.1916 −1.36816
\(396\) 0 0
\(397\) −0.421308 −0.0211449 −0.0105724 0.999944i \(-0.503365\pi\)
−0.0105724 + 0.999944i \(0.503365\pi\)
\(398\) −34.2626 −1.71743
\(399\) 0 0
\(400\) 40.7049 2.03525
\(401\) 30.3737 1.51679 0.758394 0.651796i \(-0.225984\pi\)
0.758394 + 0.651796i \(0.225984\pi\)
\(402\) 0 0
\(403\) 11.8019 0.587895
\(404\) −7.55437 −0.375844
\(405\) 0 0
\(406\) −0.716099 −0.0355394
\(407\) −30.2831 −1.50108
\(408\) 0 0
\(409\) 23.5636 1.16514 0.582572 0.812779i \(-0.302046\pi\)
0.582572 + 0.812779i \(0.302046\pi\)
\(410\) 20.3022 1.00266
\(411\) 0 0
\(412\) −82.1269 −4.04610
\(413\) 2.49495 0.122769
\(414\) 0 0
\(415\) −14.2986 −0.701890
\(416\) 59.4557 2.91505
\(417\) 0 0
\(418\) 61.2522 2.99594
\(419\) 31.9090 1.55885 0.779427 0.626493i \(-0.215510\pi\)
0.779427 + 0.626493i \(0.215510\pi\)
\(420\) 0 0
\(421\) −19.6752 −0.958911 −0.479456 0.877566i \(-0.659166\pi\)
−0.479456 + 0.877566i \(0.659166\pi\)
\(422\) 25.0854 1.22114
\(423\) 0 0
\(424\) 104.792 5.08913
\(425\) 11.5487 0.560193
\(426\) 0 0
\(427\) −2.51967 −0.121935
\(428\) 85.4340 4.12961
\(429\) 0 0
\(430\) −47.9534 −2.31252
\(431\) −34.6117 −1.66719 −0.833594 0.552378i \(-0.813720\pi\)
−0.833594 + 0.552378i \(0.813720\pi\)
\(432\) 0 0
\(433\) 27.3428 1.31401 0.657006 0.753885i \(-0.271823\pi\)
0.657006 + 0.753885i \(0.271823\pi\)
\(434\) −2.19134 −0.105188
\(435\) 0 0
\(436\) −71.3045 −3.41487
\(437\) −4.28684 −0.205067
\(438\) 0 0
\(439\) 26.0183 1.24178 0.620892 0.783896i \(-0.286771\pi\)
0.620892 + 0.783896i \(0.286771\pi\)
\(440\) −129.351 −6.16659
\(441\) 0 0
\(442\) 35.1337 1.67114
\(443\) 7.52187 0.357375 0.178687 0.983906i \(-0.442815\pi\)
0.178687 + 0.983906i \(0.442815\pi\)
\(444\) 0 0
\(445\) −18.1316 −0.859521
\(446\) −58.1312 −2.75259
\(447\) 0 0
\(448\) −4.58668 −0.216700
\(449\) −41.2802 −1.94813 −0.974067 0.226261i \(-0.927350\pi\)
−0.974067 + 0.226261i \(0.927350\pi\)
\(450\) 0 0
\(451\) −14.0579 −0.661963
\(452\) 1.25444 0.0590039
\(453\) 0 0
\(454\) −28.4665 −1.33600
\(455\) −2.99560 −0.140436
\(456\) 0 0
\(457\) 18.9955 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(458\) 25.8594 1.20833
\(459\) 0 0
\(460\) 14.8440 0.692105
\(461\) 23.7949 1.10824 0.554119 0.832437i \(-0.313055\pi\)
0.554119 + 0.832437i \(0.313055\pi\)
\(462\) 0 0
\(463\) −2.43032 −0.112947 −0.0564733 0.998404i \(-0.517986\pi\)
−0.0564733 + 0.998404i \(0.517986\pi\)
\(464\) 12.0278 0.558376
\(465\) 0 0
\(466\) −47.8350 −2.21591
\(467\) 6.85798 0.317350 0.158675 0.987331i \(-0.449278\pi\)
0.158675 + 0.987331i \(0.449278\pi\)
\(468\) 0 0
\(469\) 1.46285 0.0675481
\(470\) 94.5693 4.36216
\(471\) 0 0
\(472\) −77.6280 −3.57312
\(473\) 33.2045 1.52675
\(474\) 0 0
\(475\) −14.5077 −0.665658
\(476\) −4.69273 −0.215091
\(477\) 0 0
\(478\) 4.09298 0.187208
\(479\) −16.3926 −0.748999 −0.374500 0.927227i \(-0.622186\pi\)
−0.374500 + 0.927227i \(0.622186\pi\)
\(480\) 0 0
\(481\) 21.8207 0.994937
\(482\) 15.9595 0.726934
\(483\) 0 0
\(484\) 90.4720 4.11236
\(485\) −10.2691 −0.466297
\(486\) 0 0
\(487\) −31.4676 −1.42594 −0.712968 0.701197i \(-0.752649\pi\)
−0.712968 + 0.701197i \(0.752649\pi\)
\(488\) 78.3971 3.54887
\(489\) 0 0
\(490\) −53.5525 −2.41926
\(491\) 34.0891 1.53842 0.769209 0.638997i \(-0.220650\pi\)
0.769209 + 0.638997i \(0.220650\pi\)
\(492\) 0 0
\(493\) 3.41249 0.153691
\(494\) −44.1357 −1.98576
\(495\) 0 0
\(496\) 36.8064 1.65265
\(497\) −3.77096 −0.169151
\(498\) 0 0
\(499\) 23.1962 1.03840 0.519201 0.854652i \(-0.326229\pi\)
0.519201 + 0.854652i \(0.326229\pi\)
\(500\) −23.9844 −1.07261
\(501\) 0 0
\(502\) −17.5628 −0.783864
\(503\) 26.0785 1.16278 0.581391 0.813624i \(-0.302509\pi\)
0.581391 + 0.813624i \(0.302509\pi\)
\(504\) 0 0
\(505\) −4.26689 −0.189874
\(506\) −14.2884 −0.635198
\(507\) 0 0
\(508\) 8.73643 0.387616
\(509\) 39.1346 1.73461 0.867305 0.497777i \(-0.165850\pi\)
0.867305 + 0.497777i \(0.165850\pi\)
\(510\) 0 0
\(511\) 1.68381 0.0744875
\(512\) −15.3514 −0.678442
\(513\) 0 0
\(514\) −7.65645 −0.337711
\(515\) −46.3872 −2.04406
\(516\) 0 0
\(517\) −65.4830 −2.87994
\(518\) −4.05159 −0.178017
\(519\) 0 0
\(520\) 93.2051 4.08731
\(521\) 12.7309 0.557753 0.278876 0.960327i \(-0.410038\pi\)
0.278876 + 0.960327i \(0.410038\pi\)
\(522\) 0 0
\(523\) −29.4547 −1.28796 −0.643982 0.765041i \(-0.722719\pi\)
−0.643982 + 0.765041i \(0.722719\pi\)
\(524\) −51.8674 −2.26584
\(525\) 0 0
\(526\) −45.3700 −1.97823
\(527\) 10.4426 0.454886
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 97.0518 4.21566
\(531\) 0 0
\(532\) 5.89510 0.255585
\(533\) 10.1296 0.438760
\(534\) 0 0
\(535\) 48.2552 2.08625
\(536\) −45.5151 −1.96595
\(537\) 0 0
\(538\) 54.9546 2.36926
\(539\) 37.0815 1.59721
\(540\) 0 0
\(541\) −23.8062 −1.02351 −0.511754 0.859132i \(-0.671004\pi\)
−0.511754 + 0.859132i \(0.671004\pi\)
\(542\) 69.7981 2.99809
\(543\) 0 0
\(544\) 52.6077 2.25553
\(545\) −40.2745 −1.72517
\(546\) 0 0
\(547\) 14.3531 0.613695 0.306847 0.951759i \(-0.400726\pi\)
0.306847 + 0.951759i \(0.400726\pi\)
\(548\) −15.3729 −0.656696
\(549\) 0 0
\(550\) −48.3554 −2.06188
\(551\) −4.28684 −0.182625
\(552\) 0 0
\(553\) 2.51906 0.107121
\(554\) −14.2016 −0.603367
\(555\) 0 0
\(556\) −60.7674 −2.57711
\(557\) −18.3031 −0.775528 −0.387764 0.921759i \(-0.626752\pi\)
−0.387764 + 0.921759i \(0.626752\pi\)
\(558\) 0 0
\(559\) −23.9258 −1.01195
\(560\) −9.34229 −0.394784
\(561\) 0 0
\(562\) 28.9708 1.22206
\(563\) −4.64703 −0.195849 −0.0979245 0.995194i \(-0.531220\pi\)
−0.0979245 + 0.995194i \(0.531220\pi\)
\(564\) 0 0
\(565\) 0.708538 0.0298084
\(566\) −16.8649 −0.708883
\(567\) 0 0
\(568\) 117.330 4.92305
\(569\) 36.6541 1.53662 0.768309 0.640079i \(-0.221098\pi\)
0.768309 + 0.640079i \(0.221098\pi\)
\(570\) 0 0
\(571\) −5.18890 −0.217149 −0.108574 0.994088i \(-0.534629\pi\)
−0.108574 + 0.994088i \(0.534629\pi\)
\(572\) −105.823 −4.42470
\(573\) 0 0
\(574\) −1.88082 −0.0785040
\(575\) 3.38424 0.141132
\(576\) 0 0
\(577\) 10.5672 0.439920 0.219960 0.975509i \(-0.429407\pi\)
0.219960 + 0.975509i \(0.429407\pi\)
\(578\) −14.2952 −0.594601
\(579\) 0 0
\(580\) 14.8440 0.616363
\(581\) 1.32464 0.0549552
\(582\) 0 0
\(583\) −67.2019 −2.78322
\(584\) −52.3902 −2.16792
\(585\) 0 0
\(586\) 8.38296 0.346297
\(587\) 6.64788 0.274387 0.137194 0.990544i \(-0.456192\pi\)
0.137194 + 0.990544i \(0.456192\pi\)
\(588\) 0 0
\(589\) −13.1182 −0.540526
\(590\) −71.8945 −2.95985
\(591\) 0 0
\(592\) 68.0516 2.79690
\(593\) 31.3241 1.28633 0.643164 0.765729i \(-0.277622\pi\)
0.643164 + 0.765729i \(0.277622\pi\)
\(594\) 0 0
\(595\) −2.65057 −0.108663
\(596\) 7.23669 0.296427
\(597\) 0 0
\(598\) 10.2956 0.421020
\(599\) −19.9482 −0.815059 −0.407530 0.913192i \(-0.633610\pi\)
−0.407530 + 0.913192i \(0.633610\pi\)
\(600\) 0 0
\(601\) −16.9631 −0.691940 −0.345970 0.938246i \(-0.612450\pi\)
−0.345970 + 0.938246i \(0.612450\pi\)
\(602\) 4.44246 0.181061
\(603\) 0 0
\(604\) 21.3425 0.868413
\(605\) 51.1007 2.07754
\(606\) 0 0
\(607\) −4.19316 −0.170195 −0.0850975 0.996373i \(-0.527120\pi\)
−0.0850975 + 0.996373i \(0.527120\pi\)
\(608\) −66.0868 −2.68017
\(609\) 0 0
\(610\) 72.6068 2.93976
\(611\) 47.1842 1.90887
\(612\) 0 0
\(613\) 29.0079 1.17162 0.585809 0.810449i \(-0.300777\pi\)
0.585809 + 0.810449i \(0.300777\pi\)
\(614\) 52.7792 2.12999
\(615\) 0 0
\(616\) 11.9833 0.482820
\(617\) 32.8667 1.32316 0.661581 0.749874i \(-0.269886\pi\)
0.661581 + 0.749874i \(0.269886\pi\)
\(618\) 0 0
\(619\) −11.2241 −0.451137 −0.225568 0.974227i \(-0.572424\pi\)
−0.225568 + 0.974227i \(0.572424\pi\)
\(620\) 45.4242 1.82428
\(621\) 0 0
\(622\) 42.5014 1.70415
\(623\) 1.67973 0.0672972
\(624\) 0 0
\(625\) −30.4681 −1.21872
\(626\) −86.0505 −3.43927
\(627\) 0 0
\(628\) −91.2954 −3.64308
\(629\) 19.3074 0.769837
\(630\) 0 0
\(631\) 37.9311 1.51001 0.755007 0.655717i \(-0.227634\pi\)
0.755007 + 0.655717i \(0.227634\pi\)
\(632\) −78.3781 −3.11771
\(633\) 0 0
\(634\) 10.7723 0.427823
\(635\) 4.93454 0.195821
\(636\) 0 0
\(637\) −26.7194 −1.05866
\(638\) −14.2884 −0.565684
\(639\) 0 0
\(640\) 42.8928 1.69549
\(641\) −18.8879 −0.746028 −0.373014 0.927826i \(-0.621676\pi\)
−0.373014 + 0.927826i \(0.621676\pi\)
\(642\) 0 0
\(643\) −5.97232 −0.235525 −0.117763 0.993042i \(-0.537572\pi\)
−0.117763 + 0.993042i \(0.537572\pi\)
\(644\) −1.37516 −0.0541891
\(645\) 0 0
\(646\) −39.0522 −1.53649
\(647\) 11.1527 0.438458 0.219229 0.975673i \(-0.429646\pi\)
0.219229 + 0.975673i \(0.429646\pi\)
\(648\) 0 0
\(649\) 49.7821 1.95412
\(650\) 34.8429 1.36665
\(651\) 0 0
\(652\) 63.4530 2.48501
\(653\) 37.7960 1.47907 0.739536 0.673117i \(-0.235045\pi\)
0.739536 + 0.673117i \(0.235045\pi\)
\(654\) 0 0
\(655\) −29.2960 −1.14469
\(656\) 31.5908 1.23341
\(657\) 0 0
\(658\) −8.76101 −0.341540
\(659\) −42.4555 −1.65383 −0.826915 0.562327i \(-0.809906\pi\)
−0.826915 + 0.562327i \(0.809906\pi\)
\(660\) 0 0
\(661\) 29.4778 1.14655 0.573276 0.819362i \(-0.305672\pi\)
0.573276 + 0.819362i \(0.305672\pi\)
\(662\) 45.9882 1.78738
\(663\) 0 0
\(664\) −41.2148 −1.59945
\(665\) 3.32970 0.129120
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 35.2734 1.36477
\(669\) 0 0
\(670\) −42.1534 −1.62853
\(671\) −50.2754 −1.94086
\(672\) 0 0
\(673\) 28.5427 1.10024 0.550121 0.835085i \(-0.314582\pi\)
0.550121 + 0.835085i \(0.314582\pi\)
\(674\) −15.1371 −0.583058
\(675\) 0 0
\(676\) 9.60763 0.369524
\(677\) −13.7924 −0.530083 −0.265042 0.964237i \(-0.585386\pi\)
−0.265042 + 0.964237i \(0.585386\pi\)
\(678\) 0 0
\(679\) 0.951344 0.0365092
\(680\) 82.4699 3.16258
\(681\) 0 0
\(682\) −43.7242 −1.67428
\(683\) −1.11404 −0.0426275 −0.0213137 0.999773i \(-0.506785\pi\)
−0.0213137 + 0.999773i \(0.506785\pi\)
\(684\) 0 0
\(685\) −8.68296 −0.331759
\(686\) 9.97386 0.380804
\(687\) 0 0
\(688\) −74.6167 −2.84473
\(689\) 48.4228 1.84476
\(690\) 0 0
\(691\) −12.6753 −0.482190 −0.241095 0.970502i \(-0.577507\pi\)
−0.241095 + 0.970502i \(0.577507\pi\)
\(692\) 31.9125 1.21313
\(693\) 0 0
\(694\) −30.2766 −1.14929
\(695\) −34.3229 −1.30194
\(696\) 0 0
\(697\) 8.96285 0.339492
\(698\) −81.1243 −3.07060
\(699\) 0 0
\(700\) −4.65388 −0.175900
\(701\) −9.37223 −0.353984 −0.176992 0.984212i \(-0.556637\pi\)
−0.176992 + 0.984212i \(0.556637\pi\)
\(702\) 0 0
\(703\) −24.2543 −0.914770
\(704\) −91.5187 −3.44924
\(705\) 0 0
\(706\) −86.1292 −3.24151
\(707\) 0.395290 0.0148664
\(708\) 0 0
\(709\) −47.9902 −1.80231 −0.901155 0.433496i \(-0.857280\pi\)
−0.901155 + 0.433496i \(0.857280\pi\)
\(710\) 108.664 4.07808
\(711\) 0 0
\(712\) −52.2633 −1.95865
\(713\) 3.06011 0.114602
\(714\) 0 0
\(715\) −59.7716 −2.23533
\(716\) −115.086 −4.30095
\(717\) 0 0
\(718\) −5.44585 −0.203237
\(719\) −12.9071 −0.481354 −0.240677 0.970605i \(-0.577369\pi\)
−0.240677 + 0.970605i \(0.577369\pi\)
\(720\) 0 0
\(721\) 4.29736 0.160042
\(722\) −1.66321 −0.0618983
\(723\) 0 0
\(724\) −21.9974 −0.817529
\(725\) 3.38424 0.125687
\(726\) 0 0
\(727\) 23.6458 0.876975 0.438487 0.898737i \(-0.355514\pi\)
0.438487 + 0.898737i \(0.355514\pi\)
\(728\) −8.63463 −0.320021
\(729\) 0 0
\(730\) −48.5207 −1.79583
\(731\) −21.1700 −0.783002
\(732\) 0 0
\(733\) 12.0760 0.446036 0.223018 0.974814i \(-0.428409\pi\)
0.223018 + 0.974814i \(0.428409\pi\)
\(734\) −67.0168 −2.47363
\(735\) 0 0
\(736\) 15.4162 0.568249
\(737\) 29.1884 1.07517
\(738\) 0 0
\(739\) −17.6117 −0.647858 −0.323929 0.946081i \(-0.605004\pi\)
−0.323929 + 0.946081i \(0.605004\pi\)
\(740\) 83.9853 3.08736
\(741\) 0 0
\(742\) −8.99099 −0.330070
\(743\) −19.7219 −0.723528 −0.361764 0.932270i \(-0.617825\pi\)
−0.361764 + 0.932270i \(0.617825\pi\)
\(744\) 0 0
\(745\) 4.08746 0.149753
\(746\) 61.3933 2.24777
\(747\) 0 0
\(748\) −93.6348 −3.42363
\(749\) −4.47042 −0.163345
\(750\) 0 0
\(751\) −32.4318 −1.18345 −0.591726 0.806139i \(-0.701553\pi\)
−0.591726 + 0.806139i \(0.701553\pi\)
\(752\) 147.152 5.36609
\(753\) 0 0
\(754\) 10.2956 0.374945
\(755\) 12.0547 0.438717
\(756\) 0 0
\(757\) −3.32070 −0.120693 −0.0603464 0.998177i \(-0.519221\pi\)
−0.0603464 + 0.998177i \(0.519221\pi\)
\(758\) 79.8796 2.90136
\(759\) 0 0
\(760\) −103.600 −3.75798
\(761\) 0.490170 0.0177687 0.00888433 0.999961i \(-0.497172\pi\)
0.00888433 + 0.999961i \(0.497172\pi\)
\(762\) 0 0
\(763\) 3.73108 0.135074
\(764\) −85.7955 −3.10397
\(765\) 0 0
\(766\) 57.5273 2.07855
\(767\) −35.8709 −1.29522
\(768\) 0 0
\(769\) −41.0431 −1.48005 −0.740026 0.672578i \(-0.765187\pi\)
−0.740026 + 0.672578i \(0.765187\pi\)
\(770\) 11.0982 0.399951
\(771\) 0 0
\(772\) 83.5357 3.00652
\(773\) 19.0840 0.686405 0.343202 0.939261i \(-0.388488\pi\)
0.343202 + 0.939261i \(0.388488\pi\)
\(774\) 0 0
\(775\) 10.3561 0.372003
\(776\) −29.6002 −1.06258
\(777\) 0 0
\(778\) 48.1986 1.72800
\(779\) −11.2593 −0.403407
\(780\) 0 0
\(781\) −75.2425 −2.69239
\(782\) 9.10980 0.325766
\(783\) 0 0
\(784\) −83.3291 −2.97604
\(785\) −51.5658 −1.84046
\(786\) 0 0
\(787\) −16.4980 −0.588091 −0.294045 0.955791i \(-0.595002\pi\)
−0.294045 + 0.955791i \(0.595002\pi\)
\(788\) −105.761 −3.76759
\(789\) 0 0
\(790\) −72.5891 −2.58261
\(791\) −0.0656398 −0.00233388
\(792\) 0 0
\(793\) 36.2263 1.28643
\(794\) −1.12470 −0.0399142
\(795\) 0 0
\(796\) −65.7963 −2.33209
\(797\) 17.4840 0.619314 0.309657 0.950848i \(-0.399786\pi\)
0.309657 + 0.950848i \(0.399786\pi\)
\(798\) 0 0
\(799\) 41.7496 1.47700
\(800\) 52.1721 1.84456
\(801\) 0 0
\(802\) 81.0839 2.86317
\(803\) 33.5974 1.18563
\(804\) 0 0
\(805\) −0.776726 −0.0273760
\(806\) 31.5058 1.10974
\(807\) 0 0
\(808\) −12.2991 −0.432679
\(809\) −41.6731 −1.46515 −0.732575 0.680687i \(-0.761682\pi\)
−0.732575 + 0.680687i \(0.761682\pi\)
\(810\) 0 0
\(811\) 18.9138 0.664153 0.332076 0.943253i \(-0.392251\pi\)
0.332076 + 0.943253i \(0.392251\pi\)
\(812\) −1.37516 −0.0482588
\(813\) 0 0
\(814\) −80.8420 −2.83351
\(815\) 35.8398 1.25541
\(816\) 0 0
\(817\) 26.5942 0.930413
\(818\) 62.9041 2.19939
\(819\) 0 0
\(820\) 38.9875 1.36150
\(821\) 17.6801 0.617039 0.308520 0.951218i \(-0.400166\pi\)
0.308520 + 0.951218i \(0.400166\pi\)
\(822\) 0 0
\(823\) 32.9917 1.15002 0.575010 0.818147i \(-0.304998\pi\)
0.575010 + 0.818147i \(0.304998\pi\)
\(824\) −133.708 −4.65795
\(825\) 0 0
\(826\) 6.66039 0.231745
\(827\) −3.30724 −0.115004 −0.0575020 0.998345i \(-0.518314\pi\)
−0.0575020 + 0.998345i \(0.518314\pi\)
\(828\) 0 0
\(829\) 49.4697 1.71815 0.859077 0.511847i \(-0.171039\pi\)
0.859077 + 0.511847i \(0.171039\pi\)
\(830\) −38.1707 −1.32493
\(831\) 0 0
\(832\) 65.9445 2.28621
\(833\) −23.6419 −0.819143
\(834\) 0 0
\(835\) 19.9233 0.689473
\(836\) 117.626 4.06818
\(837\) 0 0
\(838\) 85.1824 2.94258
\(839\) 32.6645 1.12771 0.563853 0.825875i \(-0.309319\pi\)
0.563853 + 0.825875i \(0.309319\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −52.5239 −1.81009
\(843\) 0 0
\(844\) 48.1729 1.65818
\(845\) 5.42662 0.186681
\(846\) 0 0
\(847\) −4.73403 −0.162663
\(848\) 151.015 5.18588
\(849\) 0 0
\(850\) 30.8297 1.05745
\(851\) 5.65787 0.193949
\(852\) 0 0
\(853\) −25.7627 −0.882098 −0.441049 0.897483i \(-0.645393\pi\)
−0.441049 + 0.897483i \(0.645393\pi\)
\(854\) −6.72638 −0.230172
\(855\) 0 0
\(856\) 139.093 4.75409
\(857\) −35.2332 −1.20354 −0.601771 0.798669i \(-0.705538\pi\)
−0.601771 + 0.798669i \(0.705538\pi\)
\(858\) 0 0
\(859\) 30.7520 1.04924 0.524622 0.851335i \(-0.324207\pi\)
0.524622 + 0.851335i \(0.324207\pi\)
\(860\) −92.0875 −3.14016
\(861\) 0 0
\(862\) −92.3975 −3.14707
\(863\) −49.9199 −1.69929 −0.849647 0.527352i \(-0.823185\pi\)
−0.849647 + 0.527352i \(0.823185\pi\)
\(864\) 0 0
\(865\) 18.0249 0.612866
\(866\) 72.9929 2.48040
\(867\) 0 0
\(868\) −4.20815 −0.142834
\(869\) 50.2631 1.70506
\(870\) 0 0
\(871\) −21.0319 −0.712640
\(872\) −116.089 −3.93127
\(873\) 0 0
\(874\) −11.4439 −0.387096
\(875\) 1.25500 0.0424269
\(876\) 0 0
\(877\) 6.83191 0.230697 0.115349 0.993325i \(-0.463202\pi\)
0.115349 + 0.993325i \(0.463202\pi\)
\(878\) 69.4569 2.34406
\(879\) 0 0
\(880\) −186.408 −6.28382
\(881\) −32.6335 −1.09945 −0.549725 0.835346i \(-0.685268\pi\)
−0.549725 + 0.835346i \(0.685268\pi\)
\(882\) 0 0
\(883\) 39.6050 1.33282 0.666408 0.745587i \(-0.267831\pi\)
0.666408 + 0.745587i \(0.267831\pi\)
\(884\) 67.4692 2.26924
\(885\) 0 0
\(886\) 20.0800 0.674600
\(887\) −22.2862 −0.748297 −0.374148 0.927369i \(-0.622065\pi\)
−0.374148 + 0.927369i \(0.622065\pi\)
\(888\) 0 0
\(889\) −0.457142 −0.0153320
\(890\) −48.4032 −1.62248
\(891\) 0 0
\(892\) −111.632 −3.73773
\(893\) −52.4467 −1.75506
\(894\) 0 0
\(895\) −65.0031 −2.17281
\(896\) −3.97364 −0.132750
\(897\) 0 0
\(898\) −110.199 −3.67740
\(899\) 3.06011 0.102060
\(900\) 0 0
\(901\) 42.8456 1.42739
\(902\) −37.5283 −1.24956
\(903\) 0 0
\(904\) 2.04232 0.0679265
\(905\) −12.4247 −0.413010
\(906\) 0 0
\(907\) 32.7700 1.08811 0.544054 0.839050i \(-0.316888\pi\)
0.544054 + 0.839050i \(0.316888\pi\)
\(908\) −54.6657 −1.81415
\(909\) 0 0
\(910\) −7.99688 −0.265094
\(911\) 29.1155 0.964640 0.482320 0.875995i \(-0.339794\pi\)
0.482320 + 0.875995i \(0.339794\pi\)
\(912\) 0 0
\(913\) 26.4307 0.874728
\(914\) 50.7093 1.67732
\(915\) 0 0
\(916\) 49.6592 1.64079
\(917\) 2.71401 0.0896246
\(918\) 0 0
\(919\) −26.2277 −0.865173 −0.432586 0.901592i \(-0.642399\pi\)
−0.432586 + 0.901592i \(0.642399\pi\)
\(920\) 24.1671 0.796765
\(921\) 0 0
\(922\) 63.5216 2.09197
\(923\) 54.2165 1.78456
\(924\) 0 0
\(925\) 19.1476 0.629568
\(926\) −6.48786 −0.213204
\(927\) 0 0
\(928\) 15.4162 0.506062
\(929\) 49.4635 1.62285 0.811423 0.584459i \(-0.198693\pi\)
0.811423 + 0.584459i \(0.198693\pi\)
\(930\) 0 0
\(931\) 29.6994 0.973358
\(932\) −91.8602 −3.00898
\(933\) 0 0
\(934\) 18.3077 0.599046
\(935\) −52.8872 −1.72960
\(936\) 0 0
\(937\) 3.95297 0.129138 0.0645690 0.997913i \(-0.479433\pi\)
0.0645690 + 0.997913i \(0.479433\pi\)
\(938\) 3.90514 0.127507
\(939\) 0 0
\(940\) 181.607 5.92336
\(941\) 18.6695 0.608607 0.304304 0.952575i \(-0.401576\pi\)
0.304304 + 0.952575i \(0.401576\pi\)
\(942\) 0 0
\(943\) 2.62648 0.0855301
\(944\) −111.870 −3.64105
\(945\) 0 0
\(946\) 88.6410 2.88197
\(947\) −15.3063 −0.497389 −0.248694 0.968582i \(-0.580001\pi\)
−0.248694 + 0.968582i \(0.580001\pi\)
\(948\) 0 0
\(949\) −24.2088 −0.785852
\(950\) −38.7289 −1.25653
\(951\) 0 0
\(952\) −7.64011 −0.247617
\(953\) −24.5884 −0.796498 −0.398249 0.917277i \(-0.630382\pi\)
−0.398249 + 0.917277i \(0.630382\pi\)
\(954\) 0 0
\(955\) −48.4593 −1.56811
\(956\) 7.85997 0.254210
\(957\) 0 0
\(958\) −43.7609 −1.41385
\(959\) 0.804399 0.0259754
\(960\) 0 0
\(961\) −21.6357 −0.697927
\(962\) 58.2513 1.87810
\(963\) 0 0
\(964\) 30.6478 0.987100
\(965\) 47.1829 1.51887
\(966\) 0 0
\(967\) 27.3203 0.878561 0.439281 0.898350i \(-0.355233\pi\)
0.439281 + 0.898350i \(0.355233\pi\)
\(968\) 147.295 4.73424
\(969\) 0 0
\(970\) −27.4139 −0.880208
\(971\) 34.6915 1.11330 0.556652 0.830746i \(-0.312086\pi\)
0.556652 + 0.830746i \(0.312086\pi\)
\(972\) 0 0
\(973\) 3.17971 0.101937
\(974\) −84.0043 −2.69167
\(975\) 0 0
\(976\) 112.978 3.61634
\(977\) 33.7185 1.07875 0.539375 0.842065i \(-0.318660\pi\)
0.539375 + 0.842065i \(0.318660\pi\)
\(978\) 0 0
\(979\) 33.5160 1.07118
\(980\) −102.840 −3.28510
\(981\) 0 0
\(982\) 91.0024 2.90400
\(983\) 21.7136 0.692556 0.346278 0.938132i \(-0.387445\pi\)
0.346278 + 0.938132i \(0.387445\pi\)
\(984\) 0 0
\(985\) −59.7365 −1.90336
\(986\) 9.10980 0.290115
\(987\) 0 0
\(988\) −84.7562 −2.69645
\(989\) −6.20369 −0.197266
\(990\) 0 0
\(991\) −23.8122 −0.756418 −0.378209 0.925720i \(-0.623460\pi\)
−0.378209 + 0.925720i \(0.623460\pi\)
\(992\) 47.1753 1.49782
\(993\) 0 0
\(994\) −10.0668 −0.319298
\(995\) −37.1633 −1.17816
\(996\) 0 0
\(997\) −42.4682 −1.34498 −0.672490 0.740106i \(-0.734775\pi\)
−0.672490 + 0.740106i \(0.734775\pi\)
\(998\) 61.9232 1.96015
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6003.2.a.n.1.12 12
3.2 odd 2 667.2.a.b.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
667.2.a.b.1.1 12 3.2 odd 2
6003.2.a.n.1.12 12 1.1 even 1 trivial